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Let be a continuous function which increases to on an infinite interval of the form . A necessary and sufficient condition is found on a sequence increasing to which ensures that for each Dirichlet series of the form , , which is absolutely convergent in the following relation holds: where and are the maximum modulus and maximum term of the series, respectively. Bibliography: 10 titles.

We introduce generalized Dirichlet classes of analytic functions in a disc and a half-plane. We establish a relationship between these classes and their zero sets. A precise sufficient condition for a zero subset of a generalized Dirichlet class in a half-plane is obtained. Using this condition, we prove a necessary condition (which is also precise) for a system of exponential functions to be complete in the space on a half-line with regularly varying weight of order . Bibliography: 18 titles.

It is shown that each Dirichlet polynomial P of degree N which is bounded in a certain natural Euclidean norm, admits a nontrivial uniform approximation on the corresponding interval on the real axis by a Dirichlet polynomial with spectrum containing significantly fewer than N elements. Moreover, this spectrum is independent of P. Bibliography: 19 titles.

The behaviour of the sum of an entire Dirichlet series is analyzed in terms of the minimum of its modulus on a system of vertical line segments. Also a more general problem, connected with the Polya conjecture is posed and solved. It concerns the minimum modulus of an entire function with Fabri gaps and its growth along curves going to infinity. Bibliography: 33 titles.

This paper deals with analytic properties of certain Dirichlet series considered by Davenport and Heilbronn and by Titchmarsh. Bibliography: 28 titles.

Multiple Walsh series (S) on the group G{sup m} are studied. It is proved that every at most countable set is a uniqueness set for series (S) under convergence over cubes. The recovery problem is solved for the coefficients of series (S) that converge outside countable sets or outside sets of Dirichlet type. A number of analogues of the de la Vallee Poussin theorem are established for series (S). Bibliography: 28 titles.

A generalization of the well-known Levinson-Sjoberg theorem is obtained for a family of analytic functions f that have estimates of the form |f (z)| 6 M (dist(z, γ)) outside an arc γ, where M is a decreasing function on (0, ∞) that is unbounded in a neighbourhood of the origin. Applications to questions of quasianalyticity for Carleman classes are indi- cated as well as to the completeness of a system of exponentials on arcs, to analytic continuation and to representation by Dirichlet series. Bibliography: 24 titles.

Entire Dirichlet series with real coefficients are studied in the case when the sequence of sign changes of the coefficients satisfies the Levinson condition. The best possible lower estimate for the growth rate of a Dirichlet series on the positive half-axis is obtained.Bibliography: 25 titles.

It is shown that if the sequence of exponents of a Dirichlet series satisfies a Levinson-type condition, then the growth of the absolute values of its sum has a sharp lower bound on curves of bounded slope which depends only on the coefficients and the exponents of the series.

A spectral problem of Dirichlet type where is a bounded set, is considered. All the natural generalizations of the classical Weyl's spectral asymptotic formula are described. The main property of these generalizations is as follows: the leading term of the asymptotic formula is an additive function of the set .

A joint universality theorem of Voronin type is proved for general Dirichlet series under certain conditions on the coefficients and system of exponents.

The problem of colour symmetries of crystals and quasicrystals is investigated from its combinatorial point of view. For various lattices and modules in two and three dimensions, the number of colourings compatible with point and translation symmetry is given in terms of Dirichlet series generating functions.

Let, be an entire function represented in the whole of the plane by an absolutely convergent Dirichlet series such that The connection between the growth of the quantity End the behaviour of and as is described in general form.

Assuming that a conjecture of Selberg holds, along with certain other conditions, we obtain a lower bound on an interval of the critical line for the number of zeros of a function that is a linear combination of the analogues of the Riemann zeta-function that correspond to Dirichlet series of the Selberg class.

We study the action of the operators of symplectic Hecke rings of arbitrary degree on the theta-series of positive definite quadratic forms in an odd number of variables with vector-valued spherical coefficients corresponding to irreducible representations of the unitary group. We find a correspondence between generators of the Hecke rings and generalized Eichler-Brandt matrices. We apply these results to obtain conditions for linear dependence of theta-series, necessary conditions for lifting automorphic eigenforms on the orthogonal group to Siegel modular eigenforms, and an Euler expansion for symmetric Dirichlet series as a product of local zeta-functions with coefficients computed explicitly in terms of Eichler-Brandt matrices.

An approximate functional is derived for , where and are primitive Dirichlet characters modulo and , and also an approximate functional equation for an analogue of the Hardy-Selberg function. If , , then the remainder terms in these formulas are bounded by as (where is an arbitrarily small positive number).

For entire functions of finite lower Ritt order, defined by Dirichlet series with positive exponents, a criterion is found for to hold as outside some exceptional set.

A stronger version of the well-known theorem of Hayman concerning the minimum modulus of an entire function of infinite order defined by a lacunary power series is proved for series converging only in the unit disc or in a half-plane.

We investigate the spectrum of the Dirichlet Laplacian on broken strips. It is shown that there is exactly one eigenvalue below the bottom of the essential spectrum for any angle corresponding to the broken strip.

CONTENTS Introduction §1. Dirichlet forms. Dirichlet operators §2. Energy forms. Classical Dirichlet forms §3. Capacities. Quasi-regular Dirichlet forms and their regularizations §4. Sub-Markovian semigroups and diffusion processes generated by Dirichlet operators §5. Stochastic equations for processes generated by Dirichlet operators §6. Brownian motion with a drift in a Hilbert space §7. Conditions for functions from a given family to be logarithmic derivatives of a measure §8. Construction of a measure from its logarithmic derivatives §9. Stochastic representation of elements of L2(X,μ) and operations on them Conclusion References

A perfected version is proposed of the author's method for finding a lower bound for the number of zeros of combinations of Dirichlet functions that lie on intervals of the critical line.

Recently, the group of coincidence isometries of the root lattice A4 has been determined providing a classification of these isometries with respect to their coincidence indices. A more difficult task is the classification of all CSLs, since different coincidence isometries may generate the same CSL. In contrast to the typical examples in dimensions d ≤ 3, where coincidence isometries generating the same CSL can only differ by a symmetry operation, the situation is more involved in 4 dimensions. Here, we find coincidence isometries that are not related by a symmetry operation but nevertheless give rise to the same CSL. We indicate how the classification of CSLs can be obtained by making use of the icosian ring and provide explicit criteria for two isometries to generate the same CSL. Moreover, we determine the number of CSLs of a given index and encapsulate the result in a Dirichlet series generating function.

Theorems are proved on the number of zeros of linear combinations of Dirichlet L-functions that lie on intervals of the critical line.

Various types of functional equations for Dirichlet series are examined, and their equivalence is shown.

A Wiener-Hopf operator A is studied in the space of functions locally square-integrable on R and slowly increasing to ∞. The symbol of the operator is an infinitely differentiable function on R and has at ∞ a discontinuity of "vorticity point" type described either by a Blaschke function with all its zeros concentrated in a strip and bounded away from R, or by an outer function meromorphic in the complex plane with separated set of real zeros of bounded multiplicity. The operator A is one-sidedly invertible, and ind A = ±∞. Procedures are worked out for inverting it. The subspace ker A is described in terms of generalized Dirichlet series.

For a sequence of nonnegative numbers increasing to let denote the class of Dirichlet series , , absolutely convergent in . If , then let , and the central exponent. It is shown that for the relation to hold as outside some set of finite measure for each function it is necessary and sufficient that . This condition can be weakened in the case when an additional restriction is placed on the decrease of the coefficients .Bibliography: 10 titles.

This paper contains an investigation of conditions under which an analytic function represented by a Dirichlet series , (), absolutely convergent in satisfies the relation uniformly with respect to as in the complement of some sufficiently small set. The results are used to derive as simple corollaries new assertions for functions analytic in the unit disk that are represented by lacunary power series. All the assertions proved in this article are best possible or close to best possible. Bibliography: 12 titles.

Transversal vibrations of a plane rooted tree of Stieltjes strings are considered with Dirichlet boundary conditions at each pendant vertex. Continuity and Kirchhoff conditions are imposed at each interior vertex except the root. The Dirichlet problem is that with the Dirichlet condition at the root, and the Neumann problem is that with continuity and Kirchhoff conditions at the root. It is shown that strict interlacing is a sufficient condition for two sequences of real numbers to be the spectra of Neumann and Dirichlet problems generated by Stieltjes string recurrent relations on any prescribed tree.

Convolutions of Dinchlet series connected with modular forms of integral and half-integral weight are investigated. The analytic continuation of such series is constructed, and their values at critical integer points are calculated. Bibliography: 25 titles.

It was established by the author (MR 32 #5879) that any entire function of finite order may be represented in the whole plane by a Dirichlet series It is established that for suitable choice of the sequence the expression has, for large , the upper bounds1) , if has order ;2) , if has order and finite type .Bibliography: 7 titles.

An algebraicity theorem is proved for the values of convolutions of Dirichlet series connected with modular forms of various weights at lattice points of the critical strip, -adic estimates of these values are obtained.Bibliography: 24 titles.

This paper treats the questions of convergence and summability on a convex polygon of the Dirichlet series of a function which is analytic in and continuous on . Necessary and sufficient conditions for convergence are given for the case of a square; in the general case, if the necessary conditions for convergence are satisfied, it is sufficient that the integral converge.Bibliography: 7 items.

Using the Wiman-Valiron method the author studies the behavior in horizontal strips of entire functions given by Dirichlet series. Bibliography: 6 titles.

Asymptotic expansions (as e → 0) of solutions of the Dirichlet problem are obtained for a strongly elliptic equation of order 2m in a subdomain of Rn minus an e-neighborhood of a smooth submanifold, without boundary, of positive codimension. Bibliography: 18 titles.

The Euler expansion of Dirichlet series constructed from the Fourier coefficients of Siegel modular forms for principal congruence subgroups of the integral symplectic group is obtained.Bibliography: 6 titles.

A linear combination of Dirichlet -functions which are known not to have an Euler product is considered. It is proved that the interval contains for an arbitrary constant 0$ SRC=http://ej.iop.org/images/0025-5726/16/1/A04/tex_im_1296_img3.gif/> more than zeros for . Bibliography: 9 titles.

Using an isomorphism of Eichler-Shimura type, the algebraicity is proved, up to a common constant factor, of the values in the center of the critical strip of the L-series connected with a "new" modular cusp form on GA for an imaginary quadratic field twisted with arbitrary Dirichlet characters of this field.Bibliography: 8 titles.

For certain Jacquet-Langlands cusp forms over fields of CM-type it is shown that the value at s = 1 of their Dirichlet series for a certain infinite set of Hecke quasicharacters can be computed as algebraic linear combinations of a finite set of periods of a closed differential form on a real-analytic manifold with singular point.Bibliography: 5 titles.

The Dirichlet problem for Bellman's equation in a plane domain is considered. It is shown that, under certain assumptions about the type of smoothness and the weak nondegeneracy of the corresponding controlled diffusion process, the solution of this Dirichlet problem is the pay-off function of an optimal control problem. Under supplementary assumptions about smoothness, the pay-off function is also smooth.Bibliography: 11 titles.

In this paper, we study the problems of interpolation by means of entire functions of exponential type which satisfy special restrictions on their growth, and the relationship of these problems to Dirichlet series expansions in an arbitrary convex polygon of functions from a Smirnov space.Bibliography: 27 items.

For a function defined by a Dirichlet series that converges in a right half-plane, we introduce the -order in the half-plane and the -order in a half-strip . Under certain restrictions on the width of the half-strip, we obtain the inequalities , where is defined by a sequence of powers. The two extreme inequalities are sharp.Bibliography: 3 titles.